ODE与抛物型方程的输入到状态稳定化，涉及Dirichlet-Robin边界扰动
Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances
AI总结：
本文研究了在存在Dirichlet-Robin边界扰动和域内扰动的情况下，ODE与抛物型PDE级联系统的输入到状态稳定化问题，采用背向设计方法和广义Lyapunov方法进行稳定性分析。
摘要：
本文聚焦于在存在Dirichlet-Robin边界扰动和域内扰动的情况下，一个常微分方程（ODE）与抛物型偏微分方程（PDE）级联系统的输入到状态稳定化问题。对于具有Dirichlet点wise连接的级联系统，ODE的值由Robin边界条件在ODE-PDE接口处的直接输入决定，而PDE则由Dirichlet边界输入在另一端驱动。我们首先采用背向设计方法来设计边界控制器并解耦级联系统。这种解耦使得可以依次独立分析PDE和ODE系统的稳定性。然后，为了解决Dirichlet边界扰动对经典Lyapunov方法应用带来的挑战，我们利用广义Lyapunov方法，建立了包含Dirichlet边界扰动和两种其他类型扰动的级联系统的ISS性。所获得的结果表明，即使在存在不同类型扰动的情况下，仍可在Lyapunov稳定性理论框架内进行ISS分析。为了保证目标系统的适定性，我们使用提升技术和半群方法。最后，通过数值模拟来说明所提出控制方案和ISS性质在具有不同扰动的级联系统中的有效性。
英文摘要：
This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.
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